On β-Open Sets and Ideals in Topological Spaces

Authors

  • Glaisa T. Catalan
  • Roberto N. Padua
  • Michael Jr. Patula Baldado Negros Oriental State University

DOI:

https://doi.org/10.29020/nybg.ejpam.v12i3.3438

Keywords:

$\beta$-open sets, $\beta_{I}$-open sets, $\beta_{I}$-compactness, $c\beta_{I}$-compactness, $\beta_{I}$-hyperconnectedness and $c\beta_{I}$-hyperconnectednes

Abstract

Let X be a topological space and I be an ideal in X. A subset A of a topological space X is called a β-open set if A ⊆ cl(int(cl(A))). A subset A of X is called β-open with respect to the ideal I, or βI -open, if there exists an open set U such that (1) U − A ∈ I, and (2) A − cl(int(cl(U))) ∈ I. A space X is said to be a βI -compact space if it is βI -compact as a subset. An ideal topological space (X, τ, I) is said to be a cβI -compact space if it is cβI -compact as a subset. An ideal topological space (X, τ, I) is said to be a countably βI -compact space if X is countably βI -compact as a subset. Two sets A and B in an ideal topological space (X, τ, I) is said to be βI -separated if clβI (A) ∩ B = ∅ = A ∩ clβ(B). A subset A of an ideal topological space (X, τ, I) is said to be βI -connected if it cannot be expressed as a union of two βI -separated sets. An ideal topological space (X, τ, I) is said to be βI -connected if X βI -connected as a subset. In this study, we introduced the notions βI -open set, βI -compact, cβI -compact, βI -hyperconnected, cβI -hyperconnected, βI -connected and βI -separated. Moreover, we investigated the concept β-open set by determining some of its properties relative to the above-mentioned notions.

Author Biography

  • Michael Jr. Patula Baldado, Negros Oriental State University
    Dean, Graduate School

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Published

2019-07-25

Issue

Section

Nonlinear Analysis

How to Cite

On β-Open Sets and Ideals in Topological Spaces. (2019). European Journal of Pure and Applied Mathematics, 12(3), 893-905. https://doi.org/10.29020/nybg.ejpam.v12i3.3438